Search: id:A118896
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%I A118896
%S A118896 1,4,14,54,185,619,2027,6553,21044,67231,214122,680330,2158391,6840384,
%T A118896 21663503,68575557,217004842,686552743,2171766332,6869227848,
%U A118896 21725636644,68709456167,217293374285,687174291753,2173105517385,68722847672628
%N A118896 Number of powerful numbers <= 10^n.
%C A118896 These numbers agree with the asymptotic formula c*sqrt(x), with c=2.1732...(A090699). 
               - T. D. Noe (noe(AT)sspectra.com), May 09 2006
%C A118896 Filaseta & Trifonov write that a result of Bateman & Grosswald (1958) 
               implies that the asymptotic expansion of the number of powerful numbers 
               up to x is zeta(3/2)/zeta(3) * x^1/2 + zeta(2/3)/zeta(2) * x^1/3 
               + o(x^1/6). This approximates the series very closely: up to a(24), 
               all absolute errors are less than 75 and up to a(27) all are below 
               300. - Charles R Greathouse IV, Sep 23 2008
%D A118896 Michael Filaseta and Ognian Trifonov, "The distribution of squarefull 
               numbers in short intervals", Acta Arithmetica 67 (1994), pp. 323-333.
%H A118896 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PowerfulNumber.html">Powerful Number</a>
%H A118896 Charles R Greathouse IV, <a href="http://math.crg4.com/">Home Page</a> 
               [in lieu of email address]
%t A118896 nMax=10^12; lst={}; Do[lst=Join[lst, i^3 Range[Sqrt[nMax/i^3]]^2], {i,
               nMax^(1/3)}]; lst=Union[lst]; k=1; Table[While[lst[[k]]<10^n, k++ 
               ]; If[lst[[k]]==10^n, k, k-1], {n,0,12}] - T. D. Noe (noe(AT)sspectra.com), 
               May 09 2006
%o A118896 (PARI) sum(k=1,n^(1/3)+.01,if(issquarefree(k),sqrtint(n\k^3))) - Charles 
               R Greathouse IV, Sep 23 2008
%Y A118896 Cf. A001694, A090699.
%Y A118896 Sequence in context: A112872 A162482 A000651 this_sequence A145211 A060898 
               A045501
%Y A118896 Adjacent sequences: A118893 A118894 A118895 this_sequence A118897 A118898 
               A118899
%K A118896 nonn
%O A118896 0,2
%A A118896 Eric Weisstein (eric(AT)weisstein.com), May 05, 2006
%E A118896 More terms from T. D. Noe (noe(AT)sspectra.com), May 09 2006
%E A118896 Terms from a(13) on from Charles R Greathouse IV, Sep 23 2008

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